The ten-triangle conjecture for acute spherical diangles

An acute spherical diangle is a spherical diangle whose relevant angles are acute. A dissection is a subdivision into spherical triangles, and the triangles are acute when all their angles are acute.

Ten-triangle conjecture. Any dissection of an acute spherical diangle into acute spherical triangles requires at least ten triangles.

The paper proves a lower bound of nine triangles and does not establish whether ten is attainable as the minimum; proving this conjecture would strengthen the corresponding lower bound for gentile acute tetrahedra.